Reach–avoid (RA) games have significant applications in security and defense, particularly for unmanned aerial vehicles (UAVs). These problems are inherently challenging due to the need to consider obstacles and the adversarial nature of opponents, ensure optimality, and account for nonlinear dynamics. Hamilton–Jacobi (HJ) reachability analysis has emerged as a powerful tool for tackling these challenges; however, while it has been applied to games involving two spatial dimensions, directly extending this approach to three spatial dimensions is impossible due to high dimensionality. On the other hand, alternative approaches for solving RA games lack the generality to consider games with three spatial dimensions involving agents with nontrivial system dynamics. In this work, we propose a novel framework for dimensionality reduction by decomposing the problem into a horizontal RA subgame and a vertical RA subgame. We then solve each subgame using HJ reachability analysis and consider second-order dynamics that account for the defender’s acceleration. To reconstruct the solution to the original RA game from the subgames, we introduce a HJ-based tracking control algorithm in each subgame that guarantees not only capture of the attacker but also tracking of the attacker thereafter. We prove the conditions under which the capture guarantees are maintained. The effectiveness of our approach is demonstrated via numerical simulations, showing that the decomposition maintains optimality and guarantees in the original problem. Our methods are also validated in a Gazebo physics simulator, achieving successful capture of quadrotors in space with three spatial dimensions for the first time, to the best of our knowledge.
Bui et al. (Sun,) studied this question.